Theorem eqglact 19391

Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	⊢ 𝑋 = (Base‘𝐺)
eqger.r	⊢ ∼ = (𝐺 ~QG 𝑌)
eqglact.3	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
eqglact	⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

Distinct variable groups:   𝑥, +   𝑥, ∼   𝑥,𝐺   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌

Proof of Theorem eqglact

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqger.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
2		eqid 2761	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
3		eqglact.3	. . . . . . 7 ⊢ + = (+g‘𝐺)
4		eqger.r	. . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑌)
5	1, 2, 3, 4	eqgval 19389	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
6		3anass 1111	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
7	5, 6	bitrdi 290	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌))))
8	7	baibd 549	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
9	8	3impa 1127	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
10	9	abbidv 2827	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∣ 𝐴 ∼ 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
11		dfec2 8720	. . 3 ⊢ (𝐴 ∈ 𝑋 → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
12	11	3ad2ant3 1153	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
13		eqid 2761	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥))) = (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))
14	13, 1, 3, 2	grplactcnv 19253	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴))))
15	14	simprd 501	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)))
16	13, 1	grplactfval 19251	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
17	16	adantl 487	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
18	17	cnveqd 5853	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
19	1, 2	grpinvcl 19198	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
20	13, 1	grplactfval 19251	. . . . . . . 8 ⊢ (((invg‘𝐺)‘𝐴) ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
21	19, 20	syl 18	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
22	15, 18, 21	3eqtr3d 2804	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
23	22	cnveqd 5853	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
24	23	3adant2 1149	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
25	24	imaeq1d 6051	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌))
26		imacnvcnv 6207	. . 3 ⊢ (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)
27		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥))
28	27	mptpreima 6239	. . . 4 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌}
29		df-rab 3414	. . . 4 ⊢ {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
30	28, 29	eqtri 2784	. . 3 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
31	25, 26, 30	3eqtr3g 2819	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
32	10, 12, 31	3eqtr4d 2806	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739  {crab 3413   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5650   “ cima 5654  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  [cec 8715  Basecbs 17387  +_gcplusg 17428  Grpcgrp 19144  inv_gcminusg 19145   ~_QG cqg 19332

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-ec 8719  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-eqg 19335

This theorem is used by:  eqgen  19393  pzriprnglem10  21796  cldsubg  24430  tgpconncompeqg  24431  snclseqg  24435  ellcsrspsn  36406